Skip to main content

Janet-Like Gröbner Bases

  • Conference paper

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 3718)

Abstract

We define a new type of Gröbner bases called Janet-like, since their properties are similar to those for Janet bases. In particular, Janet-like bases also admit an explicit formula for the Hilbert function of polynomial ideals. Cardinality of a Janet-like basis never exceeds that of a Janet basis, but in many cases it is substantially less. Especially, Janet-like bases are much more compact than their Janet counterparts when reduced Gröbner bases have “sparce” leading monomials sets, e.g., for toric ideals. We present an algorithm for constructing Janet-like bases that is a slight modification of our Janet division algorithm. The former algorithm, by the reason of checking not more but often less number of nonmultiplicative prolongations, is more efficient than the latter one.

Keywords

  • Polynomial Ideal
  • Hilbert Function
  • Hilbert Polynomial
  • Reducible Modulo
  • Toric Ideal

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Gerdt, V.P., Blinkov, Y.A.: Janet-like monomial division. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2005. LNCS, vol. 3718, pp. 174–183. Springer, Heidelberg (2005)

    CrossRef  Google Scholar 

  2. Gerdt, V.P., Blinkov, Y.A.: Involutive Bases of Polynomial Ideals. Mathematics and Computers in Simulation 45, 519–542 (1998), http://arXiv.org/math.AC/9912027 ; Minimal Involutive Bases. Ibid., 543–560, http://arXiv.org/math.AC/9912029

  3. Gerdt, V.P.: Involutive Algorithms for Computing Gröbner Bases. In: Cojocaru, S., Pfister, G., Ufnarovski, V. (eds.) Computational Commutative and Non-Commutative algebraic geometry. NATO Science Series, pp. 199–225. IOS Press, Amsterdam (2005), http://arXiv.org/math.AC/0501111

    Google Scholar 

  4. Buchberger, B.: Gröbner Bases: an Algorithmic Method in Polynomial Ideal Theory. In: Bose, N.K. (ed.) Recent Trends in Multidimensional System Theory, Reidel, Dordrecht, pp. 184–232 (1985)

    Google Scholar 

  5. Apel, J.: Theory of Involutive Divisions and an Application to Hilbert Function Computations. Journal of Symbolic Computation 25, 683–704 (1998)

    CrossRef  MATH  MathSciNet  Google Scholar 

  6. Seiler, W.M.: Involution - The formal theory of differential equations and its applications in computer algebra and numerical analysis, Habilitation thesis, Dept. of Mathematics, University of Mannheim (2002)

    Google Scholar 

  7. Bigatti, A.M., La Scala, R., Robbiano, L.: Computing Toric Ideals. Journal of Symbolic Computation 27, 351–365 (1999)

    CrossRef  MATH  MathSciNet  Google Scholar 

  8. Pottier, L.: Computation of toric Gröbner bases, Gröbner bases of lattices and integer point sof polytopes, http://www-sop.inria.fr/safir/SAM/Bastat/doc/doc.html

  9. Gerdt, V.P., Blinkov, Y.A.: Janet Bases of Toric Ideals. In: Kredel, H., Seiler, W.K. (eds.) Proceedings of the 8th Rhine Workshop on Computer Algebra, pp. 125–135. University of Mannheim (2002), http://arXiv.org/math.AC/0501180

  10. Morales, M.: Equations des Variétés Monomiales en codimension deaux. Journal of Algebra 175, 1082–1095 (1995)

    CrossRef  MATH  MathSciNet  Google Scholar 

  11. Hemmecke, R.: Private communication

    Google Scholar 

  12. Giovinni, A., Mora, T., Niesi, G., Robbiano, L., Traverso, C.: One sugar cube, please, or selection strategies in the Buchberger algorithm. In: Proceedings of ISSAC 1991, pp. 49–54. ACM Press, New York (1991)

    CrossRef  Google Scholar 

  13. Gebauer, R., Möller, H.M.: Buchberger’s Algorithm and Staggered Linear Bases. In: Proceedings of SYMSAC 1986, pp. 218–221. ACM Press, New York (1986)

    CrossRef  Google Scholar 

  14. Möller, H.M., Mora, T., Traverso, C.: Gröbner Bases Computation Using Syzygies. In: Proceedings of ISSAC 1992, pp. 320–328. ACM Press, New York (1992)

    CrossRef  Google Scholar 

  15. Faugère, J.C.: A new efficient algorithm for computing Gröbner bases without reduction to zero (F 5). In: Proceedings of ISSAC 2002, pp. 75–83. ACM Press, New York (2002)

    CrossRef  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2005 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Gerdt, V.P., Blinkov, Y.A. (2005). Janet-Like Gröbner Bases. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2005. Lecture Notes in Computer Science, vol 3718. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11555964_16

Download citation

  • DOI: https://doi.org/10.1007/11555964_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-28966-1

  • Online ISBN: 978-3-540-32070-8

  • eBook Packages: Computer ScienceComputer Science (R0)